91Ó°ÊÓ

For the \(M / M / 1\) queue, compute (a) the expected number of arrivals during a service period and (b) the probability that no customers arrive during a service period. Hint: "Condition."

The manager of a market can hire either Mary or Alice. Mary, who gives service at an exponential rate of 20 customers per hour, can be hired at a rate of \(\$ 3\) per hour. Alice, who gives service at an exponential rate of 30 customers per hour, can be hired at a rate of \(\$ C\) per hour. The manager estimates that, on the average, each customer's time is worth \(\$ 1\) per hour and should be accounted for in the model. Assume customers arrive at a Poisson rate of 10 per hour (a) What is the average cost per hour if Mary is hired? If Alice is hired? (b) Find \(C\) if the average cost per hour is the same for Mary and Alice.

A group of \(m\) customers frequents a single-server station in the following manner. When a customer arrives, he or she either enters service if the server is free or joins the queue otherwise. Upon completing service the customer departs the system, but then returns after an exponential time with rate \(\theta\). All service times are exponentially distributed with rate \(\mu\). (a) Find the average rate at which customers enter the station. (b) Find the average time that a customer spends in the station per visit.

Consider a queueing system having two servers and no queue. There are two types of customers. Type 1 customers arrive according to a Poisson process having rate \(\lambda_{1}\), and will enter the system if either server is free. The service time of a type 1 customer is exponential with rate \(\mu_{1}\). Type 2 customers arrive according to a Poisson process having rate \(\lambda_{2}\). A type 2 customer requires the simultaneous use of both servers; hence, a type 2 arrival will only enter the system if both servers are free. The time that it takes (the two servers) to serve a type 2 customer is exponential with rate \(\mu_{2}\). Once a service is completed on a customer, that customer departs the system. (a) Define states to analyze the preceding model. (b) Give the balance equations. In terms of the solution of the balance equations, find (c) the average amount of time an entering customer spends in the system; (d) the fraction of served customers that are type \(1 .\)

Consider a sequential-service system consisting of two servers, \(A\) and \(B\). Arriving customers will enter this system only if server \(A\) is free. If a customer does enter, then he is immediately served by server \(A\). When his service by \(A\) is completed, he then goes to \(B\) if \(B\) is free, or if \(B\) is busy, he leaves the system. Upon completion of service at server \(B\), the customer departs. Assume that the (Poisson) arrival rate is two customers an hour, and that \(A\) and \(B\) serve at respective (exponential) rates of four and two customers an hour. (a) What proportion of customers enter the system? (b) What proportion of entering customers receive service from B? (c) What is the average number of customers in the system? (d) What is the average amount of time that an entering customer spends in the system?

Customers arrive at a two-server system according to a Poisson process having rate \(\lambda=5\). An arrival finding server 1 free will begin service with that server. An arrival finding server 1 busy and server 2 free will enter service with server \(2 .\) An arrival finding both servers busy goes away. Once a customer is served by either server, he departs the system. The service times at server \(i\) are exponential with rates \(\mu_{i}\), where \(\mu_{1}=4, \mu_{2}=2\) (a) What is the average time an entering customer spends in the system? (b) What proportion of time is server 2 busy?

There are two types of customers. Type 1 and 2 customers arrive in accordance with independent Poisson processes with respective rate \(\lambda_{1}\) and \(\lambda_{2}\). There are two servers. A type 1 arrival will enter service with server 1 if that server is free; if server 1 is busy and server 2 is free, then the type 1 arrival will enter service with server 2\. If both servers are busy, then the type 1 arrival will go away. A type 2 customer can only be served by server \(2 ;\) if server 2 is free when a type 2 customer arrives, then the customer enters service with that server. If server 2 is busy when a type 2 arrives, then that customer goes away. Once a customer is served by either server, he departs the system. Service times at server \(i\) are exponential with rate \(\mu_{i}\), \(i=1,2\) Suppose we want to find the average number of customers in the system. (a) Define states. (b) Give the balance equations. Do not attempt to solve them. In terms of the long-run probabilities, what is (c) the average number of customers in the system? (d) the average time a customer spends in the system?

Consider a single-server exponential system in which ordinary customers arrive at a rate \(\lambda\) and have service rate \(\mu .\) In addition, there is a special customer who has a service rate \(\mu_{1}\). Whenever this special customer arrives, she goes directly into service (if anyone else is in service, then this person is bumped back into queue). When the special customer is not being serviced, she spends an exponential amount of time (with mean \(1 / \theta\) ) out of the system. (a) What is the average arrival rate of the special customer? (b) Define an appropriate state space and set up balance equations. (c) Find the probability that an ordinary customer is bumped \(n\) times.

Consider a closed queueing network consisting of two customers moving among two servers, and suppose that after each service completion the customer is equally likely to go to either server-that is, \(P_{1,2}=P_{2,1}=\frac{1}{2}\). Let \(\mu_{i}\) denote the exponential service rate at server \(i, i=1,2\) (a) Determine the average number of customers at each server. (b) Determine the service completion rate for each server.

Customers arrive at a single-server station in accordance with a Poisson process having rate \(\lambda .\) Each customer has a value. The successive values of customers are independent and come from a uniform distribution on \((0,1)\). The service time of a customer having value \(x\) is a random variable with mean \(3+4 x\) and variance \(5 .\) (a) What is the average time a customer spends in the system? (b) What is the average time a customer having value \(x\) spends in the system?